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1!
Phenomenological Models of Neurons!
!Lecture 5!
2!
Some Linear Algebra First!!
Notes from Eero Simoncelli
3!
Vector Addition!
Notes from Eero Simoncelli
4!
Scalar Multiplication of a Vector!
5!
Vector Norm!
6!
Unit Vector!
7!
Inner Product of Vectors (Dot Product)!
Note cos θ is a measure of similarity of two vectors
8!
Outer Product of Vectors !
9!
Linear Projection!
10!
Linear Projection!
11!
Linear Projection!
12!
Linear Combinations!
13!
Vector Space!
14!
Basis Vectors!
15!
Projection using Basis Vectors!
16!
Projection using Basis Vectors!
17!
Projection using Basis Vectors!
18!
Neural encoding problem!
Notes from John Pillow
19!
Neural encoding problem!
Notes from John Pillow
20!
Naïve Approach: A Huge Look-up Table!
Notes from John Pillow
21!
Naïve Approach: A Huge Look-up Table!
Notes from John Pillow
22!
Classical Approach!
Notes from John Pillow
23!
Classical Approach: Receptive Fields!
Hubel and Weisel, 1968
24!
Classical Approach: Receptive Fields!
Notes from John Pillow Georgopolous, 1982
25!
Classical Approach: Receptive Fields!
Notes from John Pillow
• does not take time into account
26!
Modern Approach!
Notes from John Pillow
27!
Linear Models!
28!
Linear Models!
29!
Linear Models!
30!
Linear Models!
You know X (Stimulus given)
You know Y (Whether neuron
Fired or not)
Find k
31!
Finding Maxima and Minima!g Matlab demo!
32!
Vector/Matrix Calculus!
lecture notes from Dr. Xia Hong
33!
Vector/Matrix Calculus!
lecture notes from Dr. Xia Hong
34!
Vector/Matrix Calculus!
lecture notes from Dr. Xia Hong
If your notation is such that a is row vector
(some text use this notation)
!g!w
=!(aw)!w
=
a1!am
"
#
$$$$
%
&
''''= aT
35!
Vector/Matrix Calculus!
lecture notes from Dr. Xia Hong
36!
Vector/Matrix Calculus!
lecture notes from Dr. Xia Hong
37!
Vector/Matrix Calculus!
lecture notes from Dr. Xia Hong
38!
Linear Models!
You know X (Stimulus given)
You know Y (Whether neuron
Fired or not)
Find k
39!
Lease Squares Estimate!
Sn!d" kd!1
#
= Rn!1
Find k#
to min imize mean$ squared $ error E = (S " k#
$ R)2
40!
Lease Squares Estimate!
Sn!d
kd!1
"
= Rn!1
Find k"
to min imize mean# squared # error E = (S k"
# R)2
$E
$k" %
$
$k" (S k
"
# R)2
= 0
41!
Lease Squares Estimate!
Sn!d
kd!1
"
= Rn!1
Find k"
to min imize mean# squared # error E = (S k"
# R)2
$E
$k" %
$
$k" (S k
"
# R)2
= 0
%$
$k" (S k
"
# R)T (S k"
# R)&'(
)*+= 0 (AB)T = BTAT
%$
$k" (kT
"
ST # RT )(S k"
# R)&
'(
)
*+= 0
%$
$k" (kT
"
STS k"
# kT"
STR# RTS k"
+ RTR&
'(
)
*+= 0
42!
Lease Squares Estimate!
Sn!d
kd!1
"
= Rn!1
Find k"
to min imize mean# squared # error E = (S k"
# R)2
$E
$k" %
$
$k" (S k
"
# R)2
= 0
%$
$k" (kT
"
STS k"
# k1!d
T"
STd!n
Rn!1# RT
n!1Sn!d
kd!1
"
+ RTR&
'(
)
*+= 0
%$
$k" (kT
"
STS k"
# 2 Rn!1
T Sn!d
kd!1
"
+ RTR&
'(
)
*+= 0
43!
Lease Squares Estimate!Sn!d
kd!1
"
= Rn!1
Find k"
to min imize mean# squared # error E = (S k"
# R)2
$E
$k" %
$
$k" (S k
"
# R)2
= 0
%$
$k" (kT
"
STS k"
# 2RT
n!1Sn!d
kd!1
"
+ RTR&
'(
)
*+= 0
% STS k"
+ (STS)T k"
# 2STR) = 0
% 2STS k"
= 2STR
% k"
= (STS)#1STPseudo#inverse! "# $# R
d
d x! A x
!
= AT"
#$$
%
&''
A has row vectors
d
d x! x
!T
A x!
= AT x!
+ A x!"
#$$
%
&''
44!
Over determined systems!
from Wikipedia
45!
Pseudo-inverse intuition!
46!
Linear Model!
47!
Covariance!g A measure of how two variables vary
together!
48!
Linear Model!
Notes from John Pillow
49!
An Example: Homework problem!
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 104
2
4
6
8
10
12
14
16
Time (ms)
Tria
ls 1
5Stimulus: Olfactometer Valve Turning On
Response: Sensory Neuron Firing
50!0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 104
2
4
6
8
10
12
14
16
Time (ms)
Tria
ls 1
5Stimulus: Olfactometer Valve Turning On
Response: Sensory Neuron Firing
Populate Matrices S and R!2 sec stimulus history
100 ms time bins
51!
Populate Matrices S and R!
S R (# spikes)
52!
Best Linear Filter Model!
2101
0.5
0
0.5
1
1.5
2
Lag (Time in Seconds)
Lag (Time in Seconds)
Wei
ghtin
g of
the
Stim
ulus
!
" k#
= (STS)$1STPseudo$ inverse! " # $ # R
53!
Predicted Response Vs Actual Response!
0 100 200 300 400 500 600 700 800 9001
0
1
2
3
4
5
6
7
8
9
Actual Response in different time binsPredicted Response
54!
2D – Flickering Bars!
55!
Spike Triggered Average!
56!
Spike Triggered Average!
57!
Spike Triggered Average!
58!
Spike Triggered Average!
59!
Spike Triggered Average!
60!
Spike Triggered Average!
61!
Spike Triggered Average!
62!
Spike Triggered Average!
63!
Spike Triggered Average!
64!
Spike Triggered Average!
65!
Spike Triggered Average!
66!
Spike Triggered Average!
67!
Spike Triggered Average!
68!
Spike Triggered Average!
69!
Back to our homework!
Spikes
No Spikes
70!
Spike Triggered Average!
21040
20
0
20
40
60
80
100
120
Lag (Time in Seconds)
Wei
ghtin
g of
the
Stim
ulus
71!
Spike Triggered Average!
P(Stim, Spikes)
P(Stimulus)
Projection along STA axis
72!
Polynomial Model!
73!
Polynomial Model!
74!
For Homework Problem:!
Spike Triggered Average Spike Triggered Covariance
75!
Linear-Nonlinear-Poisson Cascade Model !
76!
Linear-Nonlinear-Poisson Cascade Model !
Notes from John Pillow
77!
Linear-Nonlinear-Poisson Cascade Model !
Notes from John Pillow
78!
Linear-Nonlinear-Poisson Cascade Model !
79!
Linear-Nonlinear-Poisson Cascade Model !
80!
When does STA fail?!
81!
Suppressive interactions !
82!
Other Modifications!
83!
Multi-neuron GLM!
84!
Multi-neuron GLM!
JW Pillow et al. Nature 000, 1-5 (2008) doi:10.1038/nature07140!
85!
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